Which one of the following fundamental properties of an element is always an exact whole number by definition?

Introduction / Context:
This question checks understanding of basic atomic properties and their numerical nature. In particular, it contrasts atomic number, atomic mass, and macroscopic properties such as atomic radius and atomic volume. Recognising that atomic number is defined as a whole number representing the count of protons is fundamental for reading the periodic table and for distinguishing it from atomic mass, which is usually a decimal value due to isotopes.

Given Data / Assumptions:
- The options are atomic mass, atomic number, atomic radius, and atomic volume.
- The task is to identify which of these is always an exact whole number for any element.
- We assume standard definitions of these terms in school level chemistry.

Concept / Approach:
Atomic number is defined as the number of protons in the nucleus of an atom and is therefore always a positive integer. Atomic mass listed on a periodic table is typically a weighted average of the masses of all naturally occurring isotopes, resulting in a decimal value. Atomic radius and atomic volume depend on how atoms pack and interact and are measured quantities that can vary and be expressed with decimal values. Therefore, atomic number is unique in being always a whole number.

Step-by-Step Solution:
Step 1: Recall that atomic number is the count of protons in the nucleus of an atom, which must be an integer since you cannot have a fraction of a proton.
Step 2: Look at atomic mass values on a periodic table. They are often non integer, for example 35.5 for chlorine, because they represent average masses of isotopes weighted by natural abundance.
Step 3: Atomic radius is measured experimentally and depends on many factors such as bonding and coordination number. The values are not restricted to whole numbers and are expressed with decimal precision.
Step 4: Atomic volume is derived from molar volume and atomic packing and again is a measured property that can take any positive real value, not just integers.
Step 5: Since only atomic number is inherently a count of discrete particles, it is always an exact whole number and is the correct answer.

Verification / Alternative check:
If you examine a standard periodic table, the small whole number often printed above the element symbol is the atomic number. This number always increases by one from element to element, such as 1 for hydrogen, 2 for helium, and 3 for lithium. In contrast, the larger decimal number printed below the symbol is the relative atomic mass and is very often not an integer. Additionally, reference books always talk about atomic number as an integer quantity, confirming that it must be a whole number by definition.

Why Other Options Are Wrong:
- Atomic mass: It is usually a decimal because it represents the weighted average of different isotopes, so it is not always a whole number.
- Atomic radius: It is a measured length that can take any positive real value and is typically written with decimal places, not restricted to integers.
- Atomic volume: It is calculated from bulk measurements and varies with conditions, so it is also not constrained to integer values.

Common Pitfalls:
Students sometimes confuse atomic number and atomic mass because both appear near the element symbol in the periodic table. Another mistake is to think that because some elements have approximate whole number masses, atomic mass is always an integer, which is incorrect. To avoid these issues, remember that atomic number is the fundamental count of protons and defines the identity of the element, while atomic mass is a measured average that can be fractional.

Final Answer:
Atomic number is always an exact whole number, since it counts the number of protons in the nucleus of an atom.